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## G = C3×Q8×D5order 240 = 24·3·5

### Direct product of C3, Q8 and D5

Series: Derived Chief Lower central Upper central

 Derived series C1 — C10 — C3×Q8×D5
 Chief series C1 — C5 — C10 — C30 — C6×D5 — D5×C12 — C3×Q8×D5
 Lower central C5 — C10 — C3×Q8×D5
 Upper central C1 — C6 — C3×Q8

Generators and relations for C3×Q8×D5
G = < a,b,c,d,e | a3=b4=d5=e2=1, c2=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc-1=b-1, bd=db, be=eb, cd=dc, ce=ec, ede=d-1 >

Subgroups: 180 in 76 conjugacy classes, 50 normal (16 characteristic)
C1, C2, C2, C3, C4, C4, C22, C5, C6, C6, C2×C4, Q8, Q8, D5, C10, C12, C12, C2×C6, C15, C2×Q8, Dic5, C20, D10, C2×C12, C3×Q8, C3×Q8, C3×D5, C30, Dic10, C4×D5, C5×Q8, C6×Q8, C3×Dic5, C60, C6×D5, Q8×D5, C3×Dic10, D5×C12, Q8×C15, C3×Q8×D5
Quotients: C1, C2, C3, C22, C6, Q8, C23, D5, C2×C6, C2×Q8, D10, C3×Q8, C22×C6, C3×D5, C22×D5, C6×Q8, C6×D5, Q8×D5, D5×C2×C6, C3×Q8×D5

Smallest permutation representation of C3×Q8×D5
On 120 points
Generators in S120
(1 24 14)(2 25 15)(3 21 11)(4 22 12)(5 23 13)(6 26 16)(7 27 17)(8 28 18)(9 29 19)(10 30 20)(31 51 41)(32 52 42)(33 53 43)(34 54 44)(35 55 45)(36 56 46)(37 57 47)(38 58 48)(39 59 49)(40 60 50)(61 81 71)(62 82 72)(63 83 73)(64 84 74)(65 85 75)(66 86 76)(67 87 77)(68 88 78)(69 89 79)(70 90 80)(91 111 101)(92 112 102)(93 113 103)(94 114 104)(95 115 105)(96 116 106)(97 117 107)(98 118 108)(99 119 109)(100 120 110)
(1 39 9 34)(2 40 10 35)(3 36 6 31)(4 37 7 32)(5 38 8 33)(11 46 16 41)(12 47 17 42)(13 48 18 43)(14 49 19 44)(15 50 20 45)(21 56 26 51)(22 57 27 52)(23 58 28 53)(24 59 29 54)(25 60 30 55)(61 91 66 96)(62 92 67 97)(63 93 68 98)(64 94 69 99)(65 95 70 100)(71 101 76 106)(72 102 77 107)(73 103 78 108)(74 104 79 109)(75 105 80 110)(81 111 86 116)(82 112 87 117)(83 113 88 118)(84 114 89 119)(85 115 90 120)
(1 69 9 64)(2 70 10 65)(3 66 6 61)(4 67 7 62)(5 68 8 63)(11 76 16 71)(12 77 17 72)(13 78 18 73)(14 79 19 74)(15 80 20 75)(21 86 26 81)(22 87 27 82)(23 88 28 83)(24 89 29 84)(25 90 30 85)(31 96 36 91)(32 97 37 92)(33 98 38 93)(34 99 39 94)(35 100 40 95)(41 106 46 101)(42 107 47 102)(43 108 48 103)(44 109 49 104)(45 110 50 105)(51 116 56 111)(52 117 57 112)(53 118 58 113)(54 119 59 114)(55 120 60 115)
(1 2 3 4 5)(6 7 8 9 10)(11 12 13 14 15)(16 17 18 19 20)(21 22 23 24 25)(26 27 28 29 30)(31 32 33 34 35)(36 37 38 39 40)(41 42 43 44 45)(46 47 48 49 50)(51 52 53 54 55)(56 57 58 59 60)(61 62 63 64 65)(66 67 68 69 70)(71 72 73 74 75)(76 77 78 79 80)(81 82 83 84 85)(86 87 88 89 90)(91 92 93 94 95)(96 97 98 99 100)(101 102 103 104 105)(106 107 108 109 110)(111 112 113 114 115)(116 117 118 119 120)
(1 8)(2 7)(3 6)(4 10)(5 9)(11 16)(12 20)(13 19)(14 18)(15 17)(21 26)(22 30)(23 29)(24 28)(25 27)(31 36)(32 40)(33 39)(34 38)(35 37)(41 46)(42 50)(43 49)(44 48)(45 47)(51 56)(52 60)(53 59)(54 58)(55 57)(61 66)(62 70)(63 69)(64 68)(65 67)(71 76)(72 80)(73 79)(74 78)(75 77)(81 86)(82 90)(83 89)(84 88)(85 87)(91 96)(92 100)(93 99)(94 98)(95 97)(101 106)(102 110)(103 109)(104 108)(105 107)(111 116)(112 120)(113 119)(114 118)(115 117)

G:=sub<Sym(120)| (1,24,14)(2,25,15)(3,21,11)(4,22,12)(5,23,13)(6,26,16)(7,27,17)(8,28,18)(9,29,19)(10,30,20)(31,51,41)(32,52,42)(33,53,43)(34,54,44)(35,55,45)(36,56,46)(37,57,47)(38,58,48)(39,59,49)(40,60,50)(61,81,71)(62,82,72)(63,83,73)(64,84,74)(65,85,75)(66,86,76)(67,87,77)(68,88,78)(69,89,79)(70,90,80)(91,111,101)(92,112,102)(93,113,103)(94,114,104)(95,115,105)(96,116,106)(97,117,107)(98,118,108)(99,119,109)(100,120,110), (1,39,9,34)(2,40,10,35)(3,36,6,31)(4,37,7,32)(5,38,8,33)(11,46,16,41)(12,47,17,42)(13,48,18,43)(14,49,19,44)(15,50,20,45)(21,56,26,51)(22,57,27,52)(23,58,28,53)(24,59,29,54)(25,60,30,55)(61,91,66,96)(62,92,67,97)(63,93,68,98)(64,94,69,99)(65,95,70,100)(71,101,76,106)(72,102,77,107)(73,103,78,108)(74,104,79,109)(75,105,80,110)(81,111,86,116)(82,112,87,117)(83,113,88,118)(84,114,89,119)(85,115,90,120), (1,69,9,64)(2,70,10,65)(3,66,6,61)(4,67,7,62)(5,68,8,63)(11,76,16,71)(12,77,17,72)(13,78,18,73)(14,79,19,74)(15,80,20,75)(21,86,26,81)(22,87,27,82)(23,88,28,83)(24,89,29,84)(25,90,30,85)(31,96,36,91)(32,97,37,92)(33,98,38,93)(34,99,39,94)(35,100,40,95)(41,106,46,101)(42,107,47,102)(43,108,48,103)(44,109,49,104)(45,110,50,105)(51,116,56,111)(52,117,57,112)(53,118,58,113)(54,119,59,114)(55,120,60,115), (1,2,3,4,5)(6,7,8,9,10)(11,12,13,14,15)(16,17,18,19,20)(21,22,23,24,25)(26,27,28,29,30)(31,32,33,34,35)(36,37,38,39,40)(41,42,43,44,45)(46,47,48,49,50)(51,52,53,54,55)(56,57,58,59,60)(61,62,63,64,65)(66,67,68,69,70)(71,72,73,74,75)(76,77,78,79,80)(81,82,83,84,85)(86,87,88,89,90)(91,92,93,94,95)(96,97,98,99,100)(101,102,103,104,105)(106,107,108,109,110)(111,112,113,114,115)(116,117,118,119,120), (1,8)(2,7)(3,6)(4,10)(5,9)(11,16)(12,20)(13,19)(14,18)(15,17)(21,26)(22,30)(23,29)(24,28)(25,27)(31,36)(32,40)(33,39)(34,38)(35,37)(41,46)(42,50)(43,49)(44,48)(45,47)(51,56)(52,60)(53,59)(54,58)(55,57)(61,66)(62,70)(63,69)(64,68)(65,67)(71,76)(72,80)(73,79)(74,78)(75,77)(81,86)(82,90)(83,89)(84,88)(85,87)(91,96)(92,100)(93,99)(94,98)(95,97)(101,106)(102,110)(103,109)(104,108)(105,107)(111,116)(112,120)(113,119)(114,118)(115,117)>;

G:=Group( (1,24,14)(2,25,15)(3,21,11)(4,22,12)(5,23,13)(6,26,16)(7,27,17)(8,28,18)(9,29,19)(10,30,20)(31,51,41)(32,52,42)(33,53,43)(34,54,44)(35,55,45)(36,56,46)(37,57,47)(38,58,48)(39,59,49)(40,60,50)(61,81,71)(62,82,72)(63,83,73)(64,84,74)(65,85,75)(66,86,76)(67,87,77)(68,88,78)(69,89,79)(70,90,80)(91,111,101)(92,112,102)(93,113,103)(94,114,104)(95,115,105)(96,116,106)(97,117,107)(98,118,108)(99,119,109)(100,120,110), (1,39,9,34)(2,40,10,35)(3,36,6,31)(4,37,7,32)(5,38,8,33)(11,46,16,41)(12,47,17,42)(13,48,18,43)(14,49,19,44)(15,50,20,45)(21,56,26,51)(22,57,27,52)(23,58,28,53)(24,59,29,54)(25,60,30,55)(61,91,66,96)(62,92,67,97)(63,93,68,98)(64,94,69,99)(65,95,70,100)(71,101,76,106)(72,102,77,107)(73,103,78,108)(74,104,79,109)(75,105,80,110)(81,111,86,116)(82,112,87,117)(83,113,88,118)(84,114,89,119)(85,115,90,120), (1,69,9,64)(2,70,10,65)(3,66,6,61)(4,67,7,62)(5,68,8,63)(11,76,16,71)(12,77,17,72)(13,78,18,73)(14,79,19,74)(15,80,20,75)(21,86,26,81)(22,87,27,82)(23,88,28,83)(24,89,29,84)(25,90,30,85)(31,96,36,91)(32,97,37,92)(33,98,38,93)(34,99,39,94)(35,100,40,95)(41,106,46,101)(42,107,47,102)(43,108,48,103)(44,109,49,104)(45,110,50,105)(51,116,56,111)(52,117,57,112)(53,118,58,113)(54,119,59,114)(55,120,60,115), (1,2,3,4,5)(6,7,8,9,10)(11,12,13,14,15)(16,17,18,19,20)(21,22,23,24,25)(26,27,28,29,30)(31,32,33,34,35)(36,37,38,39,40)(41,42,43,44,45)(46,47,48,49,50)(51,52,53,54,55)(56,57,58,59,60)(61,62,63,64,65)(66,67,68,69,70)(71,72,73,74,75)(76,77,78,79,80)(81,82,83,84,85)(86,87,88,89,90)(91,92,93,94,95)(96,97,98,99,100)(101,102,103,104,105)(106,107,108,109,110)(111,112,113,114,115)(116,117,118,119,120), (1,8)(2,7)(3,6)(4,10)(5,9)(11,16)(12,20)(13,19)(14,18)(15,17)(21,26)(22,30)(23,29)(24,28)(25,27)(31,36)(32,40)(33,39)(34,38)(35,37)(41,46)(42,50)(43,49)(44,48)(45,47)(51,56)(52,60)(53,59)(54,58)(55,57)(61,66)(62,70)(63,69)(64,68)(65,67)(71,76)(72,80)(73,79)(74,78)(75,77)(81,86)(82,90)(83,89)(84,88)(85,87)(91,96)(92,100)(93,99)(94,98)(95,97)(101,106)(102,110)(103,109)(104,108)(105,107)(111,116)(112,120)(113,119)(114,118)(115,117) );

G=PermutationGroup([[(1,24,14),(2,25,15),(3,21,11),(4,22,12),(5,23,13),(6,26,16),(7,27,17),(8,28,18),(9,29,19),(10,30,20),(31,51,41),(32,52,42),(33,53,43),(34,54,44),(35,55,45),(36,56,46),(37,57,47),(38,58,48),(39,59,49),(40,60,50),(61,81,71),(62,82,72),(63,83,73),(64,84,74),(65,85,75),(66,86,76),(67,87,77),(68,88,78),(69,89,79),(70,90,80),(91,111,101),(92,112,102),(93,113,103),(94,114,104),(95,115,105),(96,116,106),(97,117,107),(98,118,108),(99,119,109),(100,120,110)], [(1,39,9,34),(2,40,10,35),(3,36,6,31),(4,37,7,32),(5,38,8,33),(11,46,16,41),(12,47,17,42),(13,48,18,43),(14,49,19,44),(15,50,20,45),(21,56,26,51),(22,57,27,52),(23,58,28,53),(24,59,29,54),(25,60,30,55),(61,91,66,96),(62,92,67,97),(63,93,68,98),(64,94,69,99),(65,95,70,100),(71,101,76,106),(72,102,77,107),(73,103,78,108),(74,104,79,109),(75,105,80,110),(81,111,86,116),(82,112,87,117),(83,113,88,118),(84,114,89,119),(85,115,90,120)], [(1,69,9,64),(2,70,10,65),(3,66,6,61),(4,67,7,62),(5,68,8,63),(11,76,16,71),(12,77,17,72),(13,78,18,73),(14,79,19,74),(15,80,20,75),(21,86,26,81),(22,87,27,82),(23,88,28,83),(24,89,29,84),(25,90,30,85),(31,96,36,91),(32,97,37,92),(33,98,38,93),(34,99,39,94),(35,100,40,95),(41,106,46,101),(42,107,47,102),(43,108,48,103),(44,109,49,104),(45,110,50,105),(51,116,56,111),(52,117,57,112),(53,118,58,113),(54,119,59,114),(55,120,60,115)], [(1,2,3,4,5),(6,7,8,9,10),(11,12,13,14,15),(16,17,18,19,20),(21,22,23,24,25),(26,27,28,29,30),(31,32,33,34,35),(36,37,38,39,40),(41,42,43,44,45),(46,47,48,49,50),(51,52,53,54,55),(56,57,58,59,60),(61,62,63,64,65),(66,67,68,69,70),(71,72,73,74,75),(76,77,78,79,80),(81,82,83,84,85),(86,87,88,89,90),(91,92,93,94,95),(96,97,98,99,100),(101,102,103,104,105),(106,107,108,109,110),(111,112,113,114,115),(116,117,118,119,120)], [(1,8),(2,7),(3,6),(4,10),(5,9),(11,16),(12,20),(13,19),(14,18),(15,17),(21,26),(22,30),(23,29),(24,28),(25,27),(31,36),(32,40),(33,39),(34,38),(35,37),(41,46),(42,50),(43,49),(44,48),(45,47),(51,56),(52,60),(53,59),(54,58),(55,57),(61,66),(62,70),(63,69),(64,68),(65,67),(71,76),(72,80),(73,79),(74,78),(75,77),(81,86),(82,90),(83,89),(84,88),(85,87),(91,96),(92,100),(93,99),(94,98),(95,97),(101,106),(102,110),(103,109),(104,108),(105,107),(111,116),(112,120),(113,119),(114,118),(115,117)]])

C3×Q8×D5 is a maximal subgroup of   Dic102Dic3  D12.27D10  C60.39C23  C30.33C24  D12.29D10

60 conjugacy classes

 class 1 2A 2B 2C 3A 3B 4A 4B 4C 4D 4E 4F 5A 5B 6A 6B 6C 6D 6E 6F 10A 10B 12A ··· 12F 12G ··· 12L 15A 15B 15C 15D 20A ··· 20F 30A 30B 30C 30D 60A ··· 60L order 1 2 2 2 3 3 4 4 4 4 4 4 5 5 6 6 6 6 6 6 10 10 12 ··· 12 12 ··· 12 15 15 15 15 20 ··· 20 30 30 30 30 60 ··· 60 size 1 1 5 5 1 1 2 2 2 10 10 10 2 2 1 1 5 5 5 5 2 2 2 ··· 2 10 ··· 10 2 2 2 2 4 ··· 4 2 2 2 2 4 ··· 4

60 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 4 4 type + + + + - + + - image C1 C2 C2 C2 C3 C6 C6 C6 Q8 D5 D10 C3×Q8 C3×D5 C6×D5 Q8×D5 C3×Q8×D5 kernel C3×Q8×D5 C3×Dic10 D5×C12 Q8×C15 Q8×D5 Dic10 C4×D5 C5×Q8 C3×D5 C3×Q8 C12 D5 Q8 C4 C3 C1 # reps 1 3 3 1 2 6 6 2 2 2 6 4 4 12 2 4

Matrix representation of C3×Q8×D5 in GL4(𝔽61) generated by

 1 0 0 0 0 1 0 0 0 0 13 0 0 0 0 13
,
 60 0 0 0 0 60 0 0 0 0 0 1 0 0 60 0
,
 60 0 0 0 0 60 0 0 0 0 29 16 0 0 16 32
,
 60 1 0 0 42 18 0 0 0 0 1 0 0 0 0 1
,
 60 0 0 0 42 1 0 0 0 0 60 0 0 0 0 60
G:=sub<GL(4,GF(61))| [1,0,0,0,0,1,0,0,0,0,13,0,0,0,0,13],[60,0,0,0,0,60,0,0,0,0,0,60,0,0,1,0],[60,0,0,0,0,60,0,0,0,0,29,16,0,0,16,32],[60,42,0,0,1,18,0,0,0,0,1,0,0,0,0,1],[60,42,0,0,0,1,0,0,0,0,60,0,0,0,0,60] >;

C3×Q8×D5 in GAP, Magma, Sage, TeX

C_3\times Q_8\times D_5
% in TeX

G:=Group("C3xQ8xD5");
// GroupNames label

G:=SmallGroup(240,161);
// by ID

G=gap.SmallGroup(240,161);
# by ID

G:=PCGroup([6,-2,-2,-2,-3,-2,-5,151,260,122,6917]);
// Polycyclic

G:=Group<a,b,c,d,e|a^3=b^4=d^5=e^2=1,c^2=b^2,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c^-1=b^-1,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e=d^-1>;
// generators/relations

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